Levene, Rupert and Kribs, David and Power, Stephen Charles (2017) Commutants of weighted shift directed graph operator algebras. Proceedings of the American Mathematical Society, 145. pp. 3465-3480. ISSN 0002-9939
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Abstract
We consider non-selfadjoint operator algebras $\Lalg\lambda$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs~$G$. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for $\Lalg{\lambda}$ in the case of the single vertex graph with two edges and a suitable choice of left weight function~$\lambda$.